In this work, we investigate B-c -> D-(*T-) decays in the perturbative QCD approach, based on k(T) factorization, where T denotes a light tensor meson. For all decays considered in this work, there are no contributions from factorizable emission diagrams because the emitted tensor meson cannot be generated from (axial-)vector current or (pseudo)scalar density. We find that the annihilation amplitudes are dominant in these decays due to the large Cabibbo-Kobayashi-Maskawa elements, which can be perturbatively calculated without parametrization in the pQCD approach. The branching ratios of the B-c -> D-(*K-)(2)* decays can reach 10(-5) to 10(-4), which is one order of magnitude larger than that of the corresponding B-c -> D-(*K-)* decays, where heavy cancellation occurred between the penguin emission and tree annihilation diagrams. With such large branching ratios, they can very likely be observed in the ongoing LHCb experiments. We also predict a large percentage of transverse polarizations in those W-annihilation-diagram dominant B-c -> D-(*T-) decay channels. DOI: 10.1103/PhysRevD. 87.074027